Dempster Shafer Theory

Dempster-Shafer theory, also known as belief theory or evidence theory, is a mathematical framework for reasoning and decision-making under uncertainty. It provides a way to combine and manage evidence from multiple sources, even when the evidence may be uncertain or conflicting. Dempster-Shafer theory extends classical probability theory by allowing for the representation and combination of incomplete or uncertain information.

The key concepts in Dempster-Shafer theory are:

1. Basic Probability Assignment (BPA): In Dempster-Shafer theory, instead of assigning precise probabilities to events, a belief function is used to assign probabilities to sets of events. A belief function is defined over a power set of events and provides a measure of belief or uncertainty for each subset of events. The basic probability assignment assigns a belief value to each individual event and can capture the degree of evidence or support for that event.
2. Mass Function: The mass function, also known as the belief function or the basic belief assignment, is a mathematical function that describes the belief or uncertainty associated with each subset of events. It assigns a belief value to each subset of events and represents the degree of support or evidence for that subset. The mass function must satisfy certain normalization properties, such as the sum of all belief values adding up to one.
3. Combination Rule: The combination rule is used to combine belief functions from multiple sources of evidence. It determines how the belief functions are aggregated or fused to obtain an overall belief function. The Dempster’s rule of combination is a commonly used rule, which combines the belief functions by taking into account the degrees of overlap or conflict between the sources of evidence.
4. Dempster’s Rule of Combination: Dempster’s rule combines the belief functions by taking into account the degrees of conflict between the sources of evidence. It calculates the belief values for each subset of events based on the belief values assigned by each individual source. The rule involves combining the masses of evidence and then renormalizing the resulting belief function.
5. Decision Making: Dempster-Shafer theory can be used for decision-making by applying decision rules to the combined belief function. Decision rules determine how the belief values are mapped to decisions or actions. Common decision rules include maximum belief decision, where the decision is based on the subset of events with the highest belief value, and threshold decision, where a decision is made based on a predefined threshold value.

Dempster-Shafer theory provides a framework for reasoning and decision-making when dealing with uncertain or conflicting evidence. It allows for the fusion of evidence from different sources and provides a quantitative measure of belief or uncertainty for each event or hypothesis. Dempster-Shafer theory has applications in various fields, such as expert systems, decision support systems, pattern recognition, and information fusion.

Dempster – Shafer Theory (DST)

DST is a mathematical theory of evidence based on belief functions and plausible reasoning. It is used to combine separate pieces of information (evidence) to calculate the probability of an event.

DST offers an alternative to traditional probabilistic theory for the mathematical representation of uncertainty.

DST can be regarded as, a more general approach to represent uncertainty than the Bayesian approach.

Bayesian methods are sometimes inappropriate

Example :

Let A represent the proposition “Moore is attractive”. Then the axioms of probability insist that P(A) + P(¬A) = 1.

Now suppose that Andrew does not even know who “Moore” is, then

We cannot say that Andrew believes the proposition if he has no idea what it means.

Also, it is not fair to say that he disbelieves the proposition.

It would therefore be meaningful to denote Andrew’s belief B of

B(A) and B(¬A) as both being 0.

Certainty factors do not allow this.

Dempster-Shafer Model

The  idea  is    to    allocate   a  number  between 0 and 1  to indicate a degree of belief on  a proposal as in the probability framework.

However, it is not considered a probability but a belief mass. The distribution       of masses is called basic belief assignment.

In other words, in this formalism a degree of belief (referred as mass) is represented as a belief function rather than a Bayesian probability distribution.

Example: Belief assignment (continued from previous slide)

Suppose a system has five members, say five independent states, and exactly one of which is actual. If the original set is called S, | S | = 5, then

the set of all subsets (the power set) is called 2^S.

If each possible subset as a binary vector (describing any member is present or not by writing 1 or 0 ), then 2⁵ subsets are possible, ranging from the empty subset ( 0, 0, 0, 0, 0 ) to the “everything” subset ( 1, 1, 1, 1, 1 ).

The “empty” subset represents a “contradiction”, which is not true in any state, and is thus assigned a mass of one ;

The remaining masses are normalized so that their total is 1.

The “everything” subset is labeled as “unknown”; it represents the state where all elements are present one , in the sense that you cannot tell which is actual.

Belief and Plausibility

Shafer’s framework allows for belief about propositions to be represented as intervals, bounded by two values, belief (or support) and plausibility:

belief ≤ plausibility

Belief in a hypothesis is constituted by the sum of the masses of all

sets enclosed by it (i.e. the sum of the masses of all subsets of the hypothesis). It is the amount of belief that directly supports a given hypothesis at least in part, forming a lower bound.

Plausibility is 1 minus the sum of the masses of all sets whose intersectionwith the hypothesis is empty. It is an upper bound on the possibility that the hypothesis could possibly happen, up to that value, because there is only so much evidence that contradicts that hypothesis.

Example :

A proposition say  “the cat in the box is dead.”

Suppose we have belief of 0.5 and plausibility of 0.8 for the proposition.